Timeline for Motivations to study the cohomology of the moduli space of curves
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 8, 2017 at 17:39 | comment | added | Jim Bryan | @oxeimon what I said should make more sense with the wikipedia page if you set $q=e^{2\pi i \tau}$. Then the generators of the ring of quasi-modular forms, namely the Eisenstein series $E_2$, $E_4$, and $E_6$ have $q$ expansions where the coefficients are integers. I can't remember the exact formulas (my student has them) but I seem to remember that it involved $E_2^2$ and so it should be weight 4. | |
| Jul 7, 2017 at 16:35 | comment | added | Will Chen | @JimBryan Interesting... What is your definition of a quasi-modular form? (the definition on wikipedia doesn't seem to make sense with what you said). Also, do you know the level of the form? (ie, the subgroup of SL(2,Z) that stabilizes it)? | |
| Jul 7, 2017 at 14:19 | comment | added | Jim Bryan | Right, and the number of branched points is determined by the Riemann-Hurwitz formula which in this case turns out to be independent of the degree $d$ of the map $C\to E$. So the above enumerative problem is really a family of problems indexed by $d$. It turns out that $\sum_d n_d q^d$, the generating function for the solutions to that problem, is equal to a quasi-modular form! | |
| Jul 7, 2017 at 8:20 | comment | added | abx | @oxeimon: up to a finite choice, such a curve is determined by the branch divisor, which in this case consists of 2 points, up to automorphisms of $E$: this gives a one-parameter family. | |
| Jul 7, 2017 at 5:07 | comment | added | Will Chen | Naive question - How do you see that the space of genus 2 curves admitting a degree $d$ map to $E$ is 1-dimensional? (ie, why is it a 1-cycle)? | |
| Jul 7, 2017 at 3:20 | history | edited | Jim Bryan | CC BY-SA 3.0 |
added 8 characters in body
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| Jul 7, 2017 at 3:12 | history | answered | Jim Bryan | CC BY-SA 3.0 |